Studio G-Hilb

The McKay correspondence is one of my favourite sagas in algebraic geometry, partly because of the role it played in my mathematical upbringing. I went through the Dynkin classification of finite subgroups of \text{SL}_2(\mathbb{C}) via the Du Val singularities with my DRP student last semester, which is one angle of approach. Another is via equivariant Hilbert schemes. The classical Hilbert schemes parameterise certain types of subschemes of some fixed space (for instance, subschemes of \mathbb{P}^n with fixed Hilbert polynomial). For G a finite group, the equivariant version utilised here parameterises G-clusters, which are the schematic version of group orbits. This moduli space is called the G-Hilbert scheme.

I computed the minimal or crepant resolution of the Du Val singularities of type A_n (the cyclic quotient singularities of the form \frac{1}{r}(1,-1)) in a previous post using toric methods. In this note I present two more guises of this resolution, both as moduli spaces. One is the G-Hilbert scheme, and the other is as a moduli space of quiver representations of the McKay quiver of G. Both are very natural; especially the former which comes with a map to the quotient \mathbb{A}^2/G. The extraordinary thing is that these often poorly behaved moduli spaces are smooth varieties and hence supply a resolution of the quotient. They also led the way for 3-dimensional versions of the McKay correspondence, where crepant resolutions are no longer unique and where there is only a far less elegant and concise classification of finite subgroups of \text{SL}_3(\mathbb{C}), but where the G-Hilbert scheme is still a computable crepant resolution!

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